Localized states in coupled Cahn–Hilliard equations

Abstract The classical Cahn–Hilliard (CH) equation corresponds to a gradient dynamics model that describes phase decomposition in binary mixture. In the spinodal region, an initially homogeneous state spontaneously decomposes via large-scale instability into drop, hole or labyrinthine concentration patterns of typical structure length followed by continuously ongoing coarsening process. Here, we consider coupled CH two fields and show non-reciprocal (or active non-variational) coupling may induce small-scale (Turing) instability. At corresponding primary bifurcation, branch periodically patterned steady states emerges. Furthermore, there exist localized consist patches coexisting with background. branches parity-symmetric parity-asymmetric form slanted homoclinic snaking for systems conservation law. contrast structures dynamics, here, Hopf instabilities occur at sufficiently large activity, which results oscillating travelling patterns.